Related papers: The Cauchy problem for Schr\"odinger-type partial …
In this survey the contemporary results concerning supersymmetries in generalized Schr\"odinger equations are presented. Namely, position dependent mass Sch\"odinger equations are discussed as well as the equations with matrix potentials.…
In this paper, we consider the Cauchy problem for semilinear classical wave equations \begin{equation*} u_{tt}-\Delta u=|u|^{p_S(n)}\mu(|u|) \end{equation*} with the Strauss exponent $p_S(n)$ and a modulus of continuity $\mu=\mu(\tau)$,…
We establish new intrinsic Strichartz estimates for solutions of the Cauchy problem for a class of possibly degenerate Schr\"odinger equations with a real drift.
The main subject of this paper is the analysis of sequences of exponentially bounded integrated semigroups which are related to Cauchy problems \begin{equation}\label{jed} \frac{\partial}{\partial t}u(t,x)-a(D)u(t,x)=f(t,x), \quad…
In this paper we obtain generalized Gronwall type inequality using Caputo Fractional delta operator. Also we have obtained the existence of solution of Cauchy's Type problem on fractional dynamic equations using dynamic delta operator.…
We study the Cauchy problem of quasilinear Schr\"odinger equations, for which Kenig et al. (Invent Math, 2004; Adv Math, 2006) obtained large data local well-posedness by pseudo-differential techniques and viscosity methods, while Marzuola…
In this note, we generalize the nonlinearity-recovery result in [7] for classical cubic nonlinear Schr\"odinger equations to higher-order Schr\"odinger equations with a more general nonlinearity. More precisely, we consider a…
Developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering. In this paper, we explore the regularization method…
Based on the recent study on the Vlasov-Poisson-Boltzmann system with general angular cutoff potentials [3, 4], we establish in this paper the global existence of classical solutions to the Cauchy problem of the Vlasov-Poisson-Landau system…
A class of generalized Schr\"{o}dinger problems in bounded domain is studied. A complete overview of the set of solutions is provided, depending on the values assumed by parameters involved in the problem. In order to obtain the results, we…
In this paper, we study the Cauchy problem for the inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-power potential \[iu_{t} +\Delta u-c|x|^{-a}u=\pm |x|^{-b} |u|^{\sigma } u,\;\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where…
We show continuity in generalized weighted Morrey spaces of sub-linear integral operators generated by some classical integral operators and commutators. The obtained estimates are used to study global regularity of the solution of the…
We establish existence and uniqueness of generalized solutions to the initial-boundary value problem corresponding to an Euler-Bernoulli beam model from mechanics. The governing partial differential equation is of order four and involves…
In this paper, we consider the Cauchy problem for the fractional Schr\"odinger equation $i D_t^\alpha u + (-\Delta)^{\frac{\beta}{2}} u =0$ with $0<\alpha<1$, $\beta>0$. We establish the dispersive estimates for the solutions. In…
We develop a quantum algorithm for solving high-dimensional fractional Poisson equations. By applying the Caffarelli-Silvestre extension, the $d$-dimensional fractional equation is reformulated as a local partial differential equation in…
The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schr\"odinger-type operators on a half-line from the underlying Weyl-Titchmarsh function.
This paper is a direct offspring of Ref. [J. Math. Phys. 54, 072103, (2013)] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions was maid with respect to various inconsistencies and…
Schr\"odinger operators often display singularities at the origin, the Coulomb problem in atomic physics or the various matter coupling terms in the Friedmann-Robertson-Walker problem being prominent examples. For various applications it…
This work represents a first systematic attempt to create a common ground for semi-classical and time-frequency analysis. These two different areas combined together provide interesting outcomes in terms of Schr\"odinger type equations.…
An integro-differential Kolmogorov equation is considered in H\"{o}lder-type spaces defined by a scalable L\'{e}vy measure. Some properties of those spaces and estimates of the solution are derived using probabilistic representations.